Permutations Calculator
P(n,r) — order matters
Permutations with replacement count ordered arrangements where items can be repeated. Formula: n^k (n choices, k selections). Used in PIN codes, passwords, combination locks, and license plates.
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Tip: License plate formats illustrate permutations with replacement: a plate with 3 letters + 3 digits has 26³ × 10³ = 17,576,000 combinations — enough for most states' vehicle fleets.
- 1Order matters AND repetition allowed
- 2Total = n^k (n choices for each of k positions)
- 3Example: 4-digit PIN with digits 0-9: 10^4 = 10,000 possible PINs
- 4Contrast: permutations without replacement = n!/(n-k)!
3-letter code from 26 letters, repeats allowed=26³ = 17,576 codes
4-digit PIN (0–9)=10⁴ = 10,000 PINsWithout replacement: 10×9×8×7 = 5,040
| Type | Formula | n=10, k=4 |
|---|---|---|
| Permutations with replacement | n^k | 10⁴ = 10,000 |
| Permutations without replacement | n!/(n-k)! | 10!/6! = 5,040 |
| Combinations without replacement | n!/(k!(n-k)!) | C(10,4) = 210 |
| Combinations with replacement | C(n+k-1,k) | C(13,4) = 715 |
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Fun Fact
A standard combination lock (3 numbers, 0–39) actually uses permutations WITH replacement: 40³ = 64,000 combinations. A determined thief can try all combinations in about 4 hours manually.
References
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